Asymptotic Counting in Conformal Dynamical Systems.

Format:

Book

Author/Creator:

Pollicott, Mark.

Contributor:

Urbański, Mariusz.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.271

Language:

English

Subjects (All):

Conformal geometry.

Physical Description:

1 online resource (152 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former. We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These results have direct applications to a wide variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known results and proves new results. Our new approach is founded on spectral properties of complexified Ruelle- Perron-Frobenius operators and Tauberian theorems as used in classical problems of prime number theory"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Short General Introduction

1.2. Asymptotic Counting Results

1.3. Examples

1.4. Statistical Results

1.5. Historical Overview of Applications and Examples

Chapter 2. Attracting Conformal Graph Directed Markov Systems

2.1. Thermodynamic Formalism of Subshifts of Finite Type with Countable Alphabet

Preliminaries

2.2. Attracting Conformal Countable Alphabet Graph Directed Markov Systems (GDMSs) and Countable Alphabet Attracting Iterated Function Systems (IFSs)

2.3. Complex Ruelle-Perron-Frobenius Operators

Spectrum and D-Genericity

2.4. Asymptotic Results for Multipliers

Statements and First Preparations

2.5. Complex Localized Poincaré Series ᵨ

2.6. Asymptotic Results for Multipliers

Concluding of Proofs

2.7. Asymptotic Results for Diameters

Chapter 3. Parabolic Conformal Graph Directed Markov Systems

3.1. Parabolic GDMS

3.2. Poincaré's Series for \cS*, the Associated Countable Alphabet Attracting GDMS

3.3. Asymptotic Results for Multipliers

3.4. Asymptotic Results for Diameters

Chapter 4. Central Limit Theorems

4.1. Central Limit Theorems for Multipliers and Diameters: Attracting GDMSs with Invariant Measure _{ _{\cS}}

4.2. Central Limit Theorems for Multipliers and Diameters: Parabolic GDMSs with Finite Invariant Measure _{ _{\cS}}

4.3. Central Limit Theorems: Asymptotic Counting Functions for Attracting GDMSs

4.4. Central Limit Theorems: Asymptotic Counting Functions for Parabolic GDMSs

Chapter 5. Examples and Applications, I

5.1. Attracting/Expanding Conformal Dynamical Systems

5.2. Conformal Parabolic Dynamical Systems

Chapter 6. Examples and Applications, II: Kleinian Groups

6.1. Finitely Generated Classical Schottky Groups with no Tangencies.

6.2. Generalized (allowing tangencies) Classical Schottky Groups

6.3. Fuchsian Groups

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references.

ISBN:

9781470466329

1470466325

OCLC:

1264677499